Question

1. which system has exactly one solution?

a. \\(\

$$\begin{cases} y + 4x = -2 \\\\ y = -4x + 5 \\end{cases}$$

\\)

b. \\(\

$$\begin{cases} 6x + 3y = -1 \\\\ \\ 2x + y = 4 \\end{cases}$$

\\)

c. \\(\

$$\begin{cases} \\ y = 4x - 5 \\\\ y = -\\frac{1}{4}x - 5 \\end{cases}$$

\\)

d. \\(\

$$\begin{cases} \\ \\ 3x - y = 2 \\\\ y - 4 = 3(x - 2) \\end{cases}$$

\\)

Explanation:

Step1: Rewrite to slope-intercept form

For system A:
First equation: $y = -4x -2$
Second equation: $y = -4x +5$
Slopes are equal ($-4$), y-intercepts different.

Step2: Rewrite to slope-intercept form

For system B:
First equation: $3y = -6x -1 \implies y = -2x -\frac{1}{3}$
Second equation: $y = -2x +4$
Slopes are equal ($-2$), y-intercepts different.

Step3: Rewrite to slope-intercept form

For system C:
First equation: $y = 4x -5$ (slope $4$)
Second equation: $y = -\frac{1}{4}x -5$ (slope $-\frac{1}{4}$)
Slopes are not equal.

Step4: Rewrite to slope-intercept form

For system D:
First equation: $y = 3x -2$
Second equation: $y = 3x -6 +4 \implies y = 3x -2$
Slopes and y-intercepts are equal.

Step5: Determine unique solution

A linear system has exactly one solution if slopes are different (lines intersect once). Only system C has non-equal slopes.

Answer:

C.

$$\begin{cases} y = 4x - 5 \\ y = -\frac{1}{4}x - 5 \end{cases}$$